Significance of Experimental Designs in Agricultural Research. V.K. Gupta Rajender Parsad Baidya Nath Mandal

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2 Significance of Experimental Designs in Agricultural Research V.K. Gupta Rajender Parsad Baidya Nath Mandal ICAR-Indian Agricultural Statistics Research Institute Library Avenue, Pusa, New Delhi i

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4 PREFACE Experimentation is an integral component of almost every agricultural and other scientific endeavour. Properly designed experiments not only answer all the questions of the researcher but also make efficient use of available resources while doing so. Thus, careful and efficient designing of an experiment is both an art and science. It is a common phenomenon both in this country and abroad that experimenters are able to state the research problems clearly they are after, but are often unable to devise a suitable experiment because they are not sure which design to apply to conduct the experiment and even if s/ he knows the design, then how to analyse the resulting data from such an experiment is another roadblock in the process. Thus, any form of a quick handy guidance to proper designing experiments and analysing data thereof would be of immense benefit to the experimenters. The purpose of this manuscript is to expose the readers to some modern, appropriate and efficient designs actually used by the agricultural researchers in conducting their experiments and also to introduce the readers to some useful web resources for generating layout of the designs and for analysing data from designed experiments. We have taken a case study approach for describing the designs. In other words, real life situations for conducting an experiment are described first and then a suitable design for the situation is suggested. We hope that this manuscript will be immensely useful to the experimenters. Constructive criticisms and suggestions for improvements are always welcome. The authors would like to express their sincere gratitude to Dr. S. Ayyappan, Secretary, Department of Agricultural Research and Education and Director General, Indian Council of Agricultural Research who motivated us to prepare this manuscript for the benefit of researchers in agricultural sciences. We would like to convey our grateful thanks to Deputy Director General, Education Division of the ICAR for guidance in preparation of this manuscript as well as for the financial support. The authors are also thankful to the Director, ICAR-IASRI, New Delhi for his suggestions in preparation of this document. Thanks are also due to Head and all the scientists, Division of Design of Experiments at ICAR-IASRI for their support. We wish to express our sincere thanks to Mrs. Jyoti Gangwani, Assistant Chief Technical Officer, for rendering help and support in the preparation and editing of this document. V.K. Gupta Rajender Parsad New Delhi B.N. Mandal June 2015

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6 CONTENTS 1. Introduction 1 2. Incomplete block design 4 3. Resolvable block design 6 4. Augmented design Block design with factorial structure Response surface methodology Mixed orthogonal arrays (for fractional factorials) Mixture experiments Web resources Conclusion 36

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8 1. INTRODUCTION Designing an experiment is an inevitable component of every research endeavour in agricultural sciences. The data generated through designed experiments exhibit a lot of variability. The variability may be wanted and desirable, or unwanted and undesirable, but controllable in the sense that it can be accounted for. There is also some more variability, which is unwanted, undesirable and uncontrollable. The reason for its presence is unknown. For instance, the experimental units subjected to the same treatment give rise to different observations and thus create variability. These experimental units are expected to give same response, but actually the responses are different; reasons unknown. The variation may arise because of variable fertility/soil moisture/soil depth, etc. of the two plots. The statistical methodologies, in particular the theory of linear estimation and analysis of variance, enable us to partition the total variability in the data into two major components. The first major component comprises of that part of the total variability to which we can assign causes or reasons. The second component comprises of that part of the total variability to which we cannot assign any cause or reason. This part of variability is known as experimental error. This variability arises because of some factors unidentified as a source of variation. Even a careful planning of the experiment cannot ensure total elimination of this component. The observations obtained from experimental units identically treated allow the estimation of this experimental error. Ideally one should select a design that will give experimental error as small as possible. There is, though, no rule of thumb to describe what amount of experimental error is small and what amount of it can be termed as large. A popular measure of the experimental error is the percent Coefficient of Variation (CV). Generally the researcher desires the CV to be small, though there is no degree of smallness defined. A very low value of percent CV may also be viewed with suspicion and must be ascertained by using other parameters like variability across replications for all the treatments or otherwise. The explainable part of the total variability has two major components. One major component is the conditions to study or the treatments. This part of the variability is wanted or desirable. There is always a deliberate attempt on the part of the experimenter to create variability by the application of several treatments in the experiment. So treatments are one component in every designed experiment that cause variability. The other component of the explainable part of variability is the experimental 1

9 units itself. This variability is unwanted and undesirable. The factors that cause this variability are called nuisance factors. This part of the variability is accounted for by using the principle of local control. Before planning the experiment, the experimenter must have a complete knowledge about the experimental units on which the experiment would be conducted and the sources and nature of variability in the experimental units. If this variability is substantial and is not accounted for by proper designing of experiment, then this component would sit in the experimental error and make it unduly large. The end result would be a bad experiment. There could be many ways of accounting for the variability due to experimental units. The remedy depends upon the sources and nature of the factors causing variability in the experimental units. As a matter of fact, the way to account for the variability in the experimental units will dictate what type of design is to be used. One may adopt one blocking system, two blocking systems, nested blocks or nested rows and columns, etc. for accounting for the variability. Many a time, depending upon practical constraints, a naive design is the best design. Treatments in the design could be unstructured or treatments could be the levels of a single factor e.g. 10 different tree species; 6 different varieties of wheat; 8 different formulations of chemicals, 4 different grazing systems, 9 different nutrition scenarios, 4 different power levels for running a machine, etc. In this case the interest of the experimenter is to make all the possible pair wise comparisons among treatments. Treatments in the design could be structured i.e., treatments are all possible combinations of levels of several factors (factorial experiment); e.g. combinations of three levels of nitrogen, three levels of phosphorous, two levels of potash, two different spacings, two different dates of sowing making a total of 72 treatment combinations. In this case the interest of the experimenter is to estimate the main effects of the factors and interactions among factors. Once the data has been generated through the design, the model to explain the data is defined. A typical model could be response = constant + explainable part of variability + unexplainable part of variability (or error). Utilizing the fact that the explainable part of the variability has two major components viz., due to treatments and due to experimental units, the model may be rewritten as 2

10 response = constant + treatments effect + nuisance factors effect + error. Nuisance factors could be group or block effects in a block design or row effects and column effects in a row-column design or block effects and sub-blocks nested within blocks effects in a nested block design, etc. It, therefore, follows that the model is defined by the way the experiment is designed and the way the data are analyzed follows from the model defined. Thus we have the following flow chart: Treatments ] design model analysis of data. Experimental Units The data generated are then analyzed. The broad analysis of variance from a total of n observations is given below. The degrees of freedom can be appropriately found for a given design. ANOVA Source DF SS MS F-ratio Model or explainable part of variability Unexplainable part of variability or Error s SSM MSM = SSM/s Total n 1 SST n s 1 SSE MSE = SSE/ (n s 1) The analysis of variance can further be restructured as Treatments Unstructured MSM/ MSE Treatments Structured Source DF Source DF Treatments a Treatments a Nuisance factors b Main Effects a 1 Error n a b 1 All Interactions a a 1 Total n 1 Nuisance factors b Error Total n 1 n a b 1 It may be seen from above that there is no difference in the analysis if the treatments are unstructured or structured. In case the treatments are structured, then the treatment sum of squares can be further partitioned into the sum of squares due to main effects and interactions. 3

11 However, the important question to be addressed is the choice of a design for a particular experimental setting. The choice would be determined by (a) the nature of treatments (treatments structured or unstructured); (b) variability in the experimental units and the way it is accounted for; (c) questions to be answered or inference problem. The purpose of this document is to demonstrate how experiments have been designed for different experimental situations in agricultural sciences. These designs have actually been adopted by the experimenters in agricultural sciences and the data generated have been analyzed. No claim is being made for this coverage to be exhaustive. More emphasis has been laid on efforts made by IASRI, New Delhi in last years in suggesting more appropriate, modern and efficient experimental designs to the experimenters. At places some designs have been described that have a strong potential of applications in those situations where illustrated. (For detail, the readers may also like to see A Profile of Design of Experiments at IASRI, and Glimpses of Basic Research in Design of Experiments at IASRI ; in IASRI: An Era of Excellence available at Sections 2, 3 and 4 respectively highlight the usefulness of designs with unstructured treatments in conducting experiments while sections 5, 6, 7 and 8 respectively describe the usefulness of designs with structured treatments through some real life experiments. Section 9 describes various web resources that help in disseminating knowledge about designed experiments and analysis of data, which can be fruitfully used by the experimenters. 2. INCOMPLETE BLOCK DESIGN If there is only one source of variability in the experimental units, and it is not possible to have number of homogeneous experimental units equal to the number of treatments, then recourse is made to incomplete block designs. As the name itself suggests, these designs do not accommodate all the treatments in a block. The experimental units are grouped into blocks in such a way that the experimental units within a block are as homogeneous as possible and the variability between the blocks is very large. Some examples of incomplete block designs are given in the sequel. 4

12 Blocks v = 13 = b, r = 4 = k, λ = 1, n = 52 Block Block Block Block Block Block Block Block Block Block Block Block Block Here v denotes the number of treatments, b the number of blocks, r the replication of treatments, k the block size, λ the concurrences of the treatments (or number of blocks in which a pair or treatments appear together) and n the number of experimental units. This experiment is generally run as a randomized complete block (RCB) design. But a RCB design would require blocks of size 13. These would be large blocks and the experimental units within a block may not be homogeneous. On the other hand, the design suggested has blocks of size 4. There is a reduction in the block size to the order of 1/3. Obviously, small blocks are expected to have more homogeneity among units within blocks leading thereby to smaller intra block variance compared to larger (and therefore, heterogeneous) blocks where the intra block variance is expected to be large. This design, indeed, is a balanced incomplete block (BIB) design and would require 52 experimental units. However, if the experimenter wants to cut down on the total number of observations so as to economize the cost of running the experiment, an alternative but A-efficient design could be the following: 5

13 An alternate design 2 for v = 13, b = 8, k = 4, r 1 = r 3 = r 5 = r 6 = r 7 = r 10 = 3; all other treatments are replicated twice, n = 32 Block 1 Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block The randomization of treatments to the blocks in an incomplete block design involves the following steps: (a) Randomize the treatment labels; (b) Randomize the blocks; (c) Randomize the treatments within each block. Incomplete block designs like BIB designs have been adopted by All India Coordinated Research Project on Integrated Farming Systems, IIFSR, Modipuram and at several other places. But these designs are necessarily equi-replicated and have common block size. But other incomplete block designs, as described above, are also useful alternative to BIB designs because these may not be equi-replicated or have common block size and could be available for fewer experimental units. These designs have a strong potential of application in similar experimental situations and could be a good alternative to BIB designs because these lead to saving of scarce resources. A-efficient incomplete block designs are available at Design Resources Server at 3. RESOLVABLE BLOCK DESIGN In National Agricultural Research and Education System (NARES), many experiments are conducted throughout the country for improvement of crop varieties. These trials have large number of crop varieties and, therefore, require use of incomplete block designs for maintaining homogeneity of experimental units within blocks. It has also been seen that in many crop improvement programmes initial varietal trials and advanced varietal trials are generally conducted using a RCB design. The analysis of data generated from such trials reveals that the CV in many of these experiments is high and as a consequence the precision of varieties comparisons is low. This amounts to saying that the error sum of squares is unduly high compared to the sum of squares attributable 6

14 to the model. Hence, small differences among variety effects may not be detected as significant. A large number of such trials are rejected due to high CV. Incomplete block designs described in Section 2 can be used in such experiments. But from experimenters considerations, in order to demonstrate the effect of all the treatments at one place, incomplete block designs may not be an appropriate alternative. So to circumvent this problem, recourse is made to resolvable block designs. A block design is said to be resolvable if the set of b incomplete blocks can be partitioned into r groups (or replicates) containing x blocks, with the property that each treatment is assigned to one unit in each group. In other words, the x blocks in each group form a complete replication. The number of experimental units in each block is k and the number of blocks in each replicate is x = b/r. Obviously, the number of treatments is v = xk. In resolvable block designs, the between blocks within replication variation helps in reducing the experimental error (provided the experimental units within blocks are homogeneous and between blocks within replication there is heterogeneity), increasing thereby the precision of estimation of treatment contrasts of interest. Among the class of resolvable block designs, alpha (α) - designs are very useful class of designs and have now been used extensively by agricultural scientists. A special class of resolvable block designs is square lattice and rectangular lattice. These designs are, however, a particular case of alpha designs. Resolvable block designs are also nested designs in which the experimental units are divided into bigger blocks, which are complete replicate. Within each bigger block are smaller blocks, called sub-blocks, which are incomplete blocks. The treatments are allocated to the experimental units within sub-blocks of bigger block. Resolvable block designs allow performing an experiment one replication at a time. For example, in an agricultural experiment the land may be divided into a number of large areas corresponding to the replications and then each area is subdivided into blocks. These designs have been adopted for varietal trials in the NARES. These designs are also useful for the field trials with large number of treatments/crop varieties which may not always be laid out in a single location or a single season. Therefore, it is desired that variation due to location or time periods may also be controlled along with controlling within location or time period variation. This can also be handled by using resolvable block designs. Here locations or time periods may be taken as replications and the variation within a location or a time period can be taken care of by blocking. 7

15 An Initial Varietal Trial was conducted at S.K. Nagar on mustard crop with 30 entries (treatments). An alpha design in 15 blocks and block size 6 and 3 replications was recommended. The layout of the design is the following: Replication I Replication II Replication III B 1 B 2 B 3 B 4 B 5 B 1 B 2 B 3 B 4 B 5 B 1 B 2 B 3 B 4 B The analysis of variance of the data generated from an alpha design (resolvable block design) is given in the sequel. ANOVA for RCB design ANOVA for resolvable block design Source DF Source DF Treatments 29 Treatments 29 Replications 2 Blocks 14 Error 58 Replications 2 Total 89 Blocks within Replications 12 Error 46 Total 89 It may be seen from the ANOVA table that 12 degrees of freedom due to blocks within replication are taken away from the error. This is the advantage that one gets because of the nested structure. Further, since the blocks are small, it is expected that the intra block variance would also be small. In another situation an experimenter was planning to compare 40 exotic and indigenous collections / cultivars of early sown Cauliflower. The experimenter was advised to use an alpha design in 40 treatments arranged in 12 blocks of size 10 each and 3 replications with the following layout: 8

16 Replication - I Block Block Block Block Replication - II Block Block Block Block Replication - III Block Block Block Block Alpha designs were recommended for four experiments to be conducted with cotton varieties. First two of these experiments were to be conducted to morphologically characterize varieties and hybrids with respect to about 45 morphological characters both qualitative and quantitative and document the database as a part of DUS test for PVP. The parameters of the designs are: (1) v = 70; b = 20; k = 14; r = 4; (2) v = 70; b = 28; k = 10; r = 4; (3) v = 84; b = 24; k = 14; r = 4; (4) v = 28; b = 12; k = 7; r = 3; (5) v = 14; b = 6; k = 7; r = 3; Besides, alpha designs have also been recommended and adopted at 22 centres of AICRP on Rapeseed and Mustard, Bharatpur; NBPGR, New Delhi; CSK HPKV, Palampur; IARI, New Delhi; CCS HAU, Hisar; All India Sorghum Crop Improvement Project, Hyderabad. The randomization of treatments in a resolvable block deign is done in the following way: (i) Randomize the treatments, i.e., randomly allocate the treatments to treatment numbers; (ii) Randomize the replications; (iii) Randomize the blocks within each replication; (iv) Separate randomization should be done for each replication; (v) Randomize the treatments to experimental units within each block; (vi) Separate randomization should be done for each of the blocks. 9

17 Randomized layout of alpha design (number of treatments 150, block size < 10, replication < 5) is available at Design Resources Server at www. iasri.res.in/design/alpha/catalogue.htm. In 2007, a monograph on alpha design was also published at ICAR-IASRI, New Delhi. 4. AUGMENTED DESIGN A typical experimental setting could be one where the v treatments in the experiment could be assumed to be composed of two disjoint sets, (a) first set contains w treatments called test treatments or tests, (b) second set contains u treatments called control treatments or controls, u 1; w + u = v. The interest of the experimenter is in making tests versus controls comparisons. The comparisons among tests and among controls are of no consequence to the experimenter. Two types of set ups have been studied in the literature, (a) Single replication of tests; controls replicated; (b) Tests replicated; controls replicated. Designs for situation (a) are generally termed as augmented designs while designs for second situation (b) are termed as reinforced designs. Suppose that v = w + u (w tests and u controls) treatments are arranged in b blocks with block sizes as k 1, k 2,..., k b, such that k 1 + k k b = n. The analysis of variance of the data generated from such an experiment would be the following: Source ANOVA Blocks (Adjusted) b 1 Treatments (Adjusted) v 1 DF Among Tests w 1 Among Controls u 1 Tests vs Controls 1 Error n v b + 1 Total n 1 10

18 The randomization of treatments for the case when the tests and the controls are replicated is like any other block design. However, the randomization, when the tests have single replication, is tricky and the experimenters generally commit mistake in doing a proper randomization. The steps involved are the following: (a) Follow the standard randomization procedure for the known design in control treatments or check varieties; (b) Test treatments or new varieties are randomly allotted to the remaining experimental units, as in a completely randomized design; (c) If a new treatment appears more than once, assign the different entries of the treatment to a block at random with the provision that no treatment appears more than once in a block until that treatment appears once in each of the blocks. A question generally asked by the experimenters for running this experiment with single replication of tests is as to how many times each control should be replicated in each block. This question has been answered and optimum replication of controls in each block has been obtained by minimizing the information per observation. Software has been developed and is available online at Design Resources Server for generating randomized layout of an augmented design ( design/augmented Designs/home.htm). Augmented designs for both the setups when the tests have single replication and the tests can be replicated have been used immensely by the experimenters. This software has helped in generating the randomized layout of the design with optimum replication for the situation when the tests have single replication. On the other hand for the second situation when the tests are also replicated, reinforced designs have been used. For single control situation, balanced treatment incomplete block designs are useful. These designs are available at Design Resources Server at res.in/design/btib/btib.htm. The design for tests may be an alpha design or any other incomplete block design or a block design with treatments structure as factorial in nature. In the sequel are described examples for both the experimental settings. For the purpose of illustration a design is obtained using the software for number of tests, w = 54, number of controls, u = 4, number of blocks, b = 6, block size, k = 13 (9 + 4), number of plots required = 54 + (4 6) = 78. The optimum replication of controls in this example is one. The randomized layout of the design is (T# denotes the test treatment and C# denotes the control treatment): 11

19 Randomized layout of an augmented design B 1 B 2 B 3 B 4 B 5 B 6 T44 C2 C1 T54 C4 T6 T31 C3 T45 T13 T37 T17 C1 T38 C4 T9 C1 C1 C4 T36 T10 T16 T5 C3 T33 C1 C2 C1 C3 T2 C2 T21 T18 C4 C2 T25 T35 T4 T39 T49 T7 C2 T40 C4 T26 T53 T23 T14 T11 T52 T46 C3 T3 C4 C3 T42 C3 T8 T12 T43 T15 T32 T20 T48 T1 T50 T27 T24 T29 C2 T41 T34 T51 T47 T22 T28 T19 T30 Consider another example for situation (b) where the tests are also replicated. A trial is conducted with 21 new strains of Toria vis-a-vis 3 controls (checks) using an α - design for new strains and the three checks augmented in each block once. This is a reinforced alpha design. The α - design has the parameters w = 21, b = 9, r = 3 and k = 7; and the resulting design has parameters v = , b = 9, r 1 = 3, r 0 = 9, k = 10, n = 90. Replication I Replication II Replication III B 1 B 2 B 3 B 1 B 2 B 3 B 1 B 2 B A B C A B C A B C B C A B C A B C A C A B C A B C A B 12

20 Alternatively, one could have used an α - design in 24 treatments. The two options possible are (a) v = 24; b = 9; r = 3; k = 8; n = 72; (b) v = 24; b = 16; r = 4; k = 6; n = 96. But these designs are not good for this experimental setting. As stated earlier, inference problem decides what design to use. Another reinforced alpha design in 55 treatments (50 tests and 5 check varieties of Tomato) arranged in 10 blocks each of size 15 and 2 replications was recommended for Germplasm Evaluation at NBPGR, New Delhi. The replications of the check varieties are 10 each. The layout of the design recommended is as given: Replication I Replication II B 1 B 2 B 3 B 4 B 5 B 1 B 2 B 3 B 4 B Here treatments numbered 1 to 50 are the tests and treatments numbered 51 to 55 are the controls. 13

21 14 5. BLOCK DESIGN WITH FACTORIAL STRUCTURE These designs are useful in experiments where treatments are structured. These are muti-factor experiments with each factor having several levels. In the sequel we describe experimental situations, called crop sequence experiments, where these designs have been used effectively. The underlying idea of conducting crop sequence experiments is to improve cropping intensity, soil nutrients and returns to farmers. Category I experiments A crop sequence experiment comprises of two crops, one grown in kharif season followed by another grown in rabi season. Two sets of treatments are applied in succession; m treatments are applied in kharif crop and n treatments are applied in rabi crop. On application of n treatments in rabi crop, mn treatment combinations appear. The treatment structure is factorial in nature (two factors at m and n levels respectively). It is indeed possible that the m kharif and / or n rabi treatments also have factorial structure. The observations are recorded in both the seasons. The interest of the experimenter is in (a) direct effects of kharif and rabi treatments, (b) residual effects of kharif treatments, and (c) interaction between residual effects of kharif treatments and direct effects of rabi treatments. These experiments are generally conducted as: (a) Randomized Complete Block (RCB) design for m kharif treatments; (b) For application of rabi treatments, the plots of kharif experiment are subdivided into n sub-plots (number of treatments in the rabi crop); (c) Resulting design is split-plot design for rabi treatments. As in split plot designs, (a) main-plots provide residual effect of m kharif treatments; (b) sub-plots give direct effects of n rabi treatments; (c) mainplot sub-plot interaction provides the interaction of residual effect of kharif and direct effect of rabi treatments. It is assumed that residual effects persist for one season, i.e., treatments applied to kharif crop have residual effect on rabi crop and treatments applied to rabi crop have residual effect on the following kharif crop. Usually experiments are continued for two or more number of years. In second year, m kharif treatments are again applied on main plots (same randomization) and observations are recorded sub-plot wise. Sub-plots now give residual effects of rabi treatments and main-plots give direct effects of kharif treatments and the procedure is continued.

22 But this design has some limitations, viz., (a) precision of main plot treatments (representing the residual effects) is generally low, (b) each block is a complete replicate and with large number of treatment combinations it is difficult to maintain homogeneity within block, (c) because of two error components, combined analysis of experiments conducted over years is a problem particularly when the error variances are heterogeneous, and (d) identification of best treatment combination is a problem. What is the alternative? An Incomplete Block design with orthogonal factorial structure (OFS) and / or balance is an alternative. Extended Group Divisible (EGD) designs have OFS with balance. In view of practical problems one needs a resolvable solution. But the choice of design depends upon the main effects efficiency. Keeping in mind the experimenter s interest, one needs a design with full main-effects efficiency and controlled or high two-factor efficiency. The design should also be valid even when the treatments in both or any crop also have a factorial structure. A rice wheat crop sequence experiment was to be planned by the Division of Agronomy, ICAR-IARI, New Delhi. The kharif crop (rice) had five herbicidal treatments (1, 2, 3, 4, 5) and 3 replications. The rabi crop (wheat) had four herbicidal treatments (a, b, c, d) applied. An extended group divisible design (or a rectangular design) 5 4 in 10 plots per block was advised and adopted. The kharif crop rice used 5 herbicidal treatments, while the rabi crop wheat uses 4 herbicidal treatments. The layout of the design is given below: Replication I Replication II Replication III Block 1 Block 2 Block 1 Block 2 Block 1 Block 2 1 a 1 c 5 d 3 b 5 d 4 a 5 b 5 c 2 d 2 c 2 d 2 a 3 b 5 d 3 a 1 b 3 b 3 a 4 a 4 c 4 d 4 b 4 b 3 c 5 a 2 d 5 a 4 c 2 b 5 a 1 b 1 d 1 d 1 c 1 d 1 c 2 b 2 c 2 a 2 b 5 b 2 c 3 a 3 d 3 d 5 b 3 d 1 a 4 b 4 d 4 a 3 c 4 d 4 c 2 a 3 c 1 a 5 c 1 b 5 c 15

23 The analysis of variance is given below. The analysis of data using RCB design for kharif treatments and split-plot design after giving rabi treatments is also given for comparison purpose only. RCB Design (Kharif) Analysis for first year Kharif EGD Design (Kharif) Source DF Source DF Replications 2 Blocks 5 Treatments 4 Direct (Kharif) 4 Error 8 Error 50 Total 14 Total 59 Split Plot Design (Rabi) Analysis for first year Rabi EGD Design (Rabi) Source DF Source DF Replications 2 Blocks 5 Main-Plot [Residual kharif] 4 Replications 2 Error (a) 8 Blocks within Replications 3 Sub-Plot [Direct rabi] 3 Residual (kharif) 4 MP SP 12 Direct (rabi) 3 Error (b) 30 Residual * Direct 12 Total 59 Error 35 Total 59 As can be seen from the above ANOVAs, there is only one error in the EGD design used as against two error components in the ANOVA of a split-plot design. This is a clear advantage of using the suggested design. Moreover, the design adopted is an incomplete block design. That has its own advantages. 16

24 The ANOVA for the second year is the following: Analysis for second year (EGD Design) Kharif Season Rabi Season Source DF Source DF Blocks 5 Blocks 5 Replications 2 Replications 2 Blocks within replications 3 Blocks within replications 3 Residual (Rabi) 3 Residual (Kharif) 4 Direct (Kharif) 4 Direct (Rabi) 3 Residual * Direct 12 Residual * Direct 12 Error 35 Error 35 Total 59 Total 59 Using this analysis, the best treatment can be picked up easily. Motivated by the use of block designs with factorial structure by the experimenters, resolvable designs have been obtained with any number of replications that allow estimation of main effects (direct and residual effects) with full efficiency and interactions with high efficiency. The parameters of these designs are number of treatments, v = m n = f f f h h h ; n > m, 1 2 p 1 2 q and have a common factor 1 < f < m, i.e., m = g f and n = g f; number of 1 2 blocks, b = rf; replication = r; (minimum block size) k = g 1 g 2 f. It is indeed possible that some interactions are also estimated with full efficiency. Randomized layout of these designs is available at Design Resources Server ( However, when m and n are co-primes, then there is a problem of obtaining a block design with OFS. This is an open problem. An extended group divisible design in 8 plots per block has been adopted at Kanpur Centre of AICRP on CSR (now AICRP on IFS), PDFSR (now ICAR-IIFSR), Modipuram, Kanpur Centre. The kharif rice uses 4 levels of P 2 O 5, while the rabi crops use 2 doses of P 2 O 5 in each of the twosummer crops viz. Sasbenia for green manure and Green gram. 17

25 A reinforced EGD design 13 (12+1) 2 in 13 plots per block has been adopted at Division of Agronomy, IARI, New Delhi. The kharif crop rice uses13 treatments (combination of 3 doses of N from 4 different formulations and one control), while the rabi crop mustard uses 2 doses of fertilizers. A reinforced EGD design in 7 plots per block has been adopted by PDFSR (now ICAR-IIFSR), Modipuram at 32 Research Stations across the country. The treatment combinations are levels of three factors, viz., Nitrogen (40, 80 and 120 kg/ha), Phosphorous (0, 40 and 80 kg/ha) and Potassium (0 and 40 kg/ha) and one absolute control. Further, a reinforced EGD design in 12 plots per block has been adopted by Division of Agronomy, ICAR-IARI, New Delhi, in Rainfed Agriculture. The treatments are combinations of 4 levels of biofertilizers and 2 levels each of P 2 O 5 and 2 method of placement. The 8 control treatments are the combinations of 4 biofertilizers given with and without water. Category II experiments In these experiments, once again two sets of treatments are applied in succession; m treatments are applied in kharif crop and n treatments are applied in rabi crop. On application of n treatments in rabi crop, instead of mn treatment combinations only a proper subset of mn treatment combinations appear. The interest of the experimenter is in (a) direct effects of kharif and rabi treatments, and (b) residual effects of kharif treatments. The interaction between residual effects of kharif treatments and direct effects of rabi treatments is not of interest to the experimenter. An experiment was conducted at Indore centre of AICRP on Cropping Systems Research (On-Stations) to study the effect of frequency of Phosphorous (P) application for judicious use of P fertilizers in soybean wheat sequence. The treatments tried in the experiment are given in the following table: 18

26 Treatment Soybean (Kharif) Wheat (Rabi) 1 No Phosphorus No Phosphorus 2 No Phosphorus 100% P 3 50% P 50% P 4 50% P 100% P 5 100% P No Phosphorus 6 100% P 50% P 7 100% P 100% P 8 50% P + FYM 50% P + FYM 9 50% P + FYM 50% P 10 50% P + FYM + PSM 50% P + FYM + PSM There are three replications (or blocks) in the design. The observations are recorded in both the seasons. Strictly speaking, at the end of the second season this becomes bivariate data, which is ultimately converted into univariate data by creating an index of the type gross returns or returns over variable cost or energy equivalent or protein equivalent, etc. The univariate data in the form of index is then analyzed. A close look at treatment structure shows that although there are 10 treatments applied in each season, the number of distinct treatments in each season is 5 only. After giving the treatments in the second season, there are 10 distinct treatment combinations, although there should have been 25 treatment combinations. This means that only a fraction of 25 treatments have been tried in the experiment. The distinct treatment with repetitions are Distinct Treatment (Phosphorus) Kharif crop Repetitions in Rabi crop No Phosphorus T % P T % P T % P + FYM T % P + FYM + PSM T

27 For studying the direct effects of treatments applied to Soybean, the data are analyzed as a general block design with 5 treatments, 3 blocks and replications of treatments as given in the table above. For testing the residual effect of Phosphorus treatments of soybean on wheat and the direct effect of Phosphorus treatments applied to wheat, a connection is developed between the structurally incomplete row column design and designs for Category II experiments. The row-column design has the (a) replications of the original design as rows; (b) treatments applied to Soybean crop (first set of treatments) as columns; (c) cells of the array have the treatments applied to the wheat crop (second set of treatments). The following table is generated from Table of treatments: 30 Observations First Set of Treatments Replication I 1, 3 2, 3 1, 2, 3 2, 4 5 Replication II 1, 3 2, 3 1, 2, 3 2, 4 5 Replication III 1, 3 2, 3 1, 2, 3 2, 4 5 This design is disconnected and, therefore, it is not possible to answer all the questions that needed to be answered. So the choice of treatment combinations or the fraction is not proper. One may, therefore, add one more treatment combination 50% Recommended P + FYM (T 4 ) during kharif followed by 50% Recommended P + FYM + PSM (T 5 ) during rabi to the existing 10 making it 11 treatment combinations. The structure of row-column design now is the following: 33 Observations First Set of Treatments Replication I 1, 3 2, 3 1, 2, 3 2, 4, 5 5 Replication II 1, 3 2, 3 1, 2, 3 2, 4, 5 5 Replication III 1, 3 2, 3 1, 2, 3 2, 4, 5 5 This is a connected design and all the questions can be answered using this design. However, addition of a treatment combination leads to generation of 3 more observations. On the other hand, the experimenter 20

28 may not like to add observations because of cost constraint. The problem can be remedied without increasing the number of observations. The experimenter may replace the combination 50% Recommended P + FYM (T 4 ) during kharif and 50% P (T 2 ) during rabi with 50% Recommended P + FYM (T 4 ) during kharif and 50% P + FYM + PSM (T 5 ) during rabi in any one of the three replications. It is not desirable to make this change in all the replications. It can be made in any one replication as well. Thus, no additional resources are required to run the experiment. If the replacement is done in third replication only, then the structure of rowcolumn design is the following: 33 Observations First Set of Treatments Block I 1, 3 2, 3 1, 2, 3 2, 4 5 Block II 1, 3 2, 3 1, 2, 3 2, 4 5 Block III 1, 3 2, 3 1, 2, 3 4, 5 5 The design is connected and all the questions can now be answered using this design. The designs discussed in this Section are also useful for multi-stage experiments. 6. RESPONSE SURFACE METHODOLOGY Experiments under AICRP on Soil Test Crop Response correlations are conducted on a soil with a wide range of soil fertility in terms of available nitrogen (N), phosphorus (P) and potassium (K). For getting wide ranges of soil fertility, normally fertility gradients are created in the previous season. The whole area is divided into four equal strips. On each strip four different fertilizer treatments viz. 0X, 0.5X, X and 2X are applied. Here X denotes the recommended dose of N, P and K. This is followed by sowing of an exhaust crop, preferably a crop that is not to be taken as a test crop in the next season. Demarcation of strips is maintained after harvest of the exhaust crop so as to facilitate the laying out of the soil test crop response correlation experiment in next season. The main objectives of the soil test crop response experiment are (a) to establish relationship between resultant crop yield and soil test values 21

29

30 In this design 1, 2, 3 denote, respectively the three levels of N, P, K. Points were included for studying dilution behaviour of N, P and K. However, some agronomists may feel that P and K should not be used with 0 level of N. Points 1, 2, 3 enable to study the response to N at middle levels of P, K. Similarly, points 2, 4, 5 enable to study the response to P at middle levels of N and K and points 2, 6, 7 enable to study the response to K at middle levels of N and P. This design has been adopted by all the co-ordinating centres of AICRP on STCR. This experiment also aimed at evolving basis for conjoint use of organic manures (OM) and bio-fertilizers (BF) efficiently in providing integrated nutrient supply system. For 3 levels of OM and 3 strips, the design recommended is the following: Strip 1 Strip 2 Strip 3 OM1 A B C OM2 B C A OM3 C A B Here A, B and C are 8 design points each with following structure: A B C and and and Any 5 points from 1 15 Any of the 5 points from the remaining 10 points (from 1 15) Remaining 5 points This is a semi-latin square type of arrangement. All treatment combinations are tried on each level of OM. All treatment combinations are also tried on all the strips. All the three groups viz., A, B, C appear with every level of OM and also in all the strips precisely once. This design may alternatively be viewed as reinforced resolvable block design with 3 resolvable groups. Each group is a complete replicate. The 3 levels of OM are the 3 replications or the 3 resolvable groups. There are 3 blocks within each replication. The 3 strips on each level of OM are the 3 blocks. In all there are 9 blocks. There are 7 treatment combinations in each block. Each block is reinforced with a control treatment. 23

31 Designs for fitting response surfaces have been used in different situations in agricultural research. It may indeed be possible that the doses of levels of all the factors are generally equispaced. In that case the designing of an experiment is quite involved. In the sequel we shall describe designs that have actually been used in some situations: To obtain the optimum combination of levels of the factors given in the table below (osmotic dehydration of banana), an experiment was conducted at the Division of Agricultural Engineering, ICAR-IARI, New Delhi. Sl. No. Factor Levels 1. Concentration of sugar solution 40%, 50%, 60%, 70% and 80% 2. Solution to sample ratio 1:1, 3:1, 5:1, 7:1 and 9:1 3. Temperature of osmosis 25 0 c, 35 0 c, 45 0 c, 55 0 c and 65 0 c A modified second order rotatable response surface design for 3 factors each at 5 equi-spaced levels in 36 design points was recommended. The design points with levels coded as 2, 1, 0, 1, 2 and factors coded as X1, X2 and X3 are given in the table below: No. X1 X2 X3 No. X1 X2 X3 No. X1 X2 X An experiment was conducted to study production of low fat meat products using fat replacers at Division of Biotechnology, ICAR-IVRI, Izatnagar. 24

32 The experimenter was interested in obtaining optimum combination of fat replacers. The factors and the levels tried in the experiment were: Sl. No. Factor Levels 1. Whey protein 2%, 3%, 4%, 5% and 6% 2. Guar Gum 0.50%, 0.75%, 1.00%, 1.25% and 1.50% 3. Starch 2.0%, 2.5%, 3.0%, 3.5% and 4.0% A modified second order rotatable response surface design for 3 factors, each at 5 equi-spaced levels in 28 design points was recommended. The 28 design points were essentially the same as that of design points 1-28 in the above experiment on banana. An experiment was conducted with an objective to obtain optimum storability conditions of Instant Pigeon Pea (Dal) at the Division of Agricultural Engineering, ICAR-IARI, New Delhi. The factors and the levels tried in the experiment were: Sl. No. Factor Levels 1. Soaking solution concentration 0.5, 1.0, 1.5% 2. Cooking time 8, 10, 12 minutes 3. Flaking thickness 0.5, 0.75, 1.0 mm A modified second order response surface design (Box-Behnken design with 4 center points, the number of center points decided on the basis of the modified second order response surface design) for 3 factors each at 3 equi-spaced levels in 16 design points was recommended. The design points with coded levels as 1, 0, 1 are given in the table below: X1 X2 X3 X1 X2 X

33 The Division of Post Harvest Technology, ICAR-IARI, New Delhi planned an experiment related to osmotic dehydration of Aonla with the objective to obtain optimum combination of levels of solution to sample ratio, concentration of sugar solution, revolutions per minute and temperature of osmosis. The factors and levels tried in the experiment were: Sl. No. Factor Levels 1. Solution to sample ratio 1:1, 2:1, 3:1, 4:1 and 5:1 2. Concentration of sugar solution 30%, 40%, 50%, 60% and 70% 3. Revolutions per minute 100, 150, 200, 250 and Temperature of osmosis 25 0 C, 30 0 C, 35 0 C, 40 0 C and 45 0 C A second order rotatable response surface design with orthogonal blocking for 4 factors each at 5 equi-spaced levels in 30 design points arranged in three blocks each of size 10 was suggested. The layout of design with levels coded as 2, 1, 0, 1, 2 and factors coded as X1, X2, X3, X4 is as given below: Block - I Block - II Block - III X1 X2 X3 X4 X1 X2 X3 X4 X1 X2 X3 X

34 7. MIXED ORTHOGONAL ARRAYS (FOR FRACTIONAL FACTORIALS) Orthogonal arrays (or mixed orthogonal arrays) of strength two are fractional factorial designs for a symmetric (or asymmetric) factorial experiment that permit orthogonal estimation of general mean and all main effects under the assumption that all interactions are absent. Mixed orthogonal arrays have been recommended and used in an experiment at ICAR-IARI, New Delhi. A Scientist from Division of Agricultural Chemicals, ICAR-IARI, New Delhi, wished to conduct an experiment for preparation of super absorbent composites with maximum water absorption characteristics and enhanced stability and moisture absorption behaviour in plant growth media with an aim to achieve maximum and fast rate of absorbency utilizing minimum possible concentration of monomer, cross linker and alkali. The various factors along with their levels were Factor Levels Factor Levels Nature of alkali (A) 3 Monomer concentration (F) 5 Duration of alkali exposure (B) 3 Cross linker concentration (G) Temperature of reaction (C) 3 Initiator concentration (H) 5 Concentration of alkali (D) 5 Volume of water (I) 5 Backbone: Clay ratio (E) 5 5 This is a typical asymmetric factorial experiment and even a single replication would require 421,875 runs (or design points or treatment combinations). No experimenter can afford so many runs for an experiment. The interest of the experimenter was to identify the important factors out of the 9 factors in the experiment and so main effects of the factors were of major concern. The two factor and other interactions were not of much interest and can be assumed to be absent. Obviously, one has to look for a fractional factorial or a subset of 421,875 design runs. On discussion with the experimenter it was found that only design runs could be affordable. Mixed orthogonal arrays of strength two can be used as a design for this experiment. This would allow the estimation of general mean and all main 27

35 effects without correlations. But the factors have 3 and 5 levels and these numbers are co-prime. A mixed orthogonal array of strength two for a experiment would need at least 225 runs. But it is not sure that such an array is existent. The number of design points would be another issue even if such an array existsed. But it was required to suggest to the experimenter a fractional factorial plan in 72 runs arranged in 4 blocks each of size 18 for a factorial experiment. The blocking was required because it would not be possible to maintain homogeneity among the experimental units. Blocking also adds to the complexity of the problem. However, a mixed orthogonal array of strength two exists for a factorial experiment in 72 runs. The desired fractional factorial design for a factorial experiment can be obtained from a mixed orthogonal array of strength two by collapsing one of the levels of all the factors with 6 levels. The resulting design thus obtained is a fractional factorial design for a experiment in 72 runs with four blocks of size 18 each. This, however, is not an orthogonal array. The use of this fractional factorial ensures inter effect orthogonality, but the intra effect orthogonality was not possible. This was the small cost the experimenter had to bear. The suggested design had unequal replications of the levels of factors with five levels. This was not acceptable to the experimenter. So in the following year the experimenter changed the levels of the factors to 3 and 6 respectively. With this change, a design was advised with 72 design runs only and four blocks of size 18 each. But with this change the experimenter was able to accommodate more than 9 factors in the experiment; in fact 13 factors, 7 factors with 3 levels and 6 factors with 6 levels. The new factors and levels tried in the experiment were: Factor Levels Factor Levels A (Clay type) 6 H (Temperature of reaction-1) B (Backbone clay ratio) 6 I (Nature of alkali) 3 C (Monomer concentration) D (Crosslinker concentration) 6 J (Duration of exposure) 6 6 K (Concentration of alkali) 6 E (Initiator concentration) 3 L (Temperature of reaction-2) 3 F (Volume of water) 6 M (Volume of alkali) 6 G (Duration of reaction-1)